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Topic 9 Motion in Fields 9.2

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Topic 9 Motion in Fields 9.2

1. 1. MechanicsTopic 9.2 Gravitational field, potential and energy
2. 2. Gravitational Force and FieldNewton proposed that a force of attractionexists between any two masses.This force law applies to point masses notextended massesHowever the interaction between twospherical masses is the same as if themasses were concentrated at the centres ofthe spheres.
3. 3. Newton´s Law of Gravitation Newton proposed that “every particle of matter in the universeattracts every other particle with a forcewhich is directly proportional to the productof their masses, and inversely proportionalto the square of their distance apart”
4. 4. This can be written asF = G m1m 2 r2Where G is Newton´s constant ofUniversal GravitationIt has a value of 6.667 x 10-11 Nm2kg-2m is massr distance between centre of massesF is the gravitational force
5. 5. ExampleWhat is the gravitational force betweenthe Earth and the moon?Mm =0.01230*MeMe =5.976 x 1024 kgR = 384400 km
6. 6. Gravitational Field StrengthA mass M creates a gravitational field inspace around it.If a mass m is placed at some point inspace around the mass M it willexperience the existance of the field inthe form of a gravitational force
7. 7. We define the gravitational fieldstrength as the ratio of the force themass m would experience to the mass,mThat is the gravitational field strength ata point, it is the force exerted per unitmass on a particle of small mass placedat that point
8. 8. The force experienced by a mass mplaced a distance r from a mass M isF = G Mm r2And so the gravitational field strength ofthe mass M isg=GM r2
9. 9. The units of gravitational field strengthare N kg-1The gravitational field strength is avector quantity whose direction is givenby the direction of the force a masswould experience if placed at the pointof interest
10. 10. Field Strength at the Surface of a PlanetIf we replace the particle M with asphere of mass M and radius R thenrelying on the fact that the spherebehaves as a point mass situated at itscentre the field strength at the surfaceof the sphere will be given byg= GM R2
11. 11. If the sphere is the Earth then we haveg = G Me Re2But the field strength is equal to theacceleration that is produced on the mass,hence we have that the acceleration of freefall at the surface of the Earth, gg = G Me Re2Use data to verify that g = 9.81 ms -2
12. 12. Gravitational Energy and PotentialWe know that the graviational potentialenergy increases as a mass is raised abovethe EarthThe work done in moving a mass betweentwo points is positive when moving away fromthe EarthBy definition the gravitational potential energyis taken as being zero at infinityIt is a scalar quantity
13. 13. The gravitational potential at any point in theEarth´s field is given by the formulaV = - G Me rWhere r is the distance from the centre of theEarth (providing r >R)The negative sign allows for the fact that allthe potentials are negative as they have toincrease to zero
14. 14. DefinitionThe potential is therefore a measure of theamount of work that has to be done to moveparticles between points in a gravitationalfield and its units are J kg –1The work done is independent of the pathtaken between the two points in the field, as itis the difference between the initial and finalpotentials that give the value
15. 15. Gravitational potential and strengthThe gravitational field strength is a vector quantitywhose direction is given by the direction of the forcea mass would experience if placed at the point ofinterestg= GM r2Compare this to gravitational potentialV=-GM rAnd we get the relationship that g = -V r
16. 16. EquipotentialsEquipotentials join points of equalpotential togetherThey are always perpendicular to fieldlinesThey are very simple for radial anduniform fields
17. 17. In this image the lines are equallyspaced…it is a uniform fieldIn the real world the lines are surfaces,but we cant show that on paper very well
18. 18. Equipotentials for 2 point masses is like two positive charges
19. 19. Escape SpeedThe escape speed is the speedrequired for a projectile to leave theEarth´s gravitational attraction.i.e. To get to infinity!
20. 20. If the potential at the Earth´s surface isV = - G Me ReThen the Ep change to get to infinity isG Me x m ReWhere m is the mass of the projectile
21. 21. For this amount of energy to be gained theprojectile must have had an equal amount ofEkTherefore ½mv2 = G Me x m Rev = √( 2GMe Re )But using the fact that g = G Me Re2Then v = √(2gR e )
22. 22. Find the gravitational potential 1000km above the Earth’s surface? A satellite of mass 50kg moves from a point where the potential is –20 MJkg-1 to another point where the potential is –60 MJkg-1 What is the change in potential What is the speed of the satelliteA 2000kg spacecraft in orbit at R above the Earth of radius R. The potential at the Earths surface is –60 MJ kg-1. What is the change in potential energy if the spacecraft returns to Earth
23. 23. GraphsGravitational field Gravitational potentialstrength versus distance versus distanceg α 1/r2 V α -1/r